Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 3

The $\mathcal{N}=2$ supersymmetric Calogero–Sutherland model and its eigenfunctions

Pages: 543 – 590



L. Alarie-Vézina (Département de physique, de génie physique et d’optique, Université Laval, Québec, Canada)

L. Lapointe (Instituto de Matemática y Física, Universidad de Talca, Chile)

P. Mathieu (Département de physique, de génie physique et d’optique, Université Laval, Québec, Canada)


In a recent work, we have initiated the theory of $\mathcal{N}=2$ symmetric superpolynomials. As far as the classical bases are concerned, this is a rather straightforward generalization of the $\mathcal{N}=1$ case. However this construction could not be generalized to the formulation of Jack superpolynomials. The origin of this obstruction is unraveled here, opening the path for building the desired Jack extension. Those are shown to be obtained from the non-symmetric Jack polynomials by a suitable symmetrization procedure and an appropriate dressing by the anticommuting variables. This construction is substantiated by the characterization of the $\mathcal{N}=2$ Jack superpolynomials as the eigenfunctions of the $\mathcal{N}=2$ supersymmetric version of the Calogero–Sutherland model, for which, as a side result, we demonstrate the complete integrability by displaying the explicit form of four towers of mutually commuting (bosonic) conserved quantities. The $\mathcal{N}=2$ Jack superpolynomials are orthogonal with respect to the analytical scalar product (induced by the quantum-mechanical formulation) as well as a new combinatorial scalar product defined on a suitable deformation of the power-sum basis.

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Published 21 March 2022